MATH V1101x or y Calculus I 3 pts. Prerequisites: see Courses for First-Year Students. Functions, limits, derivatives, introduction to integrals. The Help Room on the 3rd floor of Milbank Hall (Barnard College) is open during the day, Monday through Friday, to students seeking individual help from the instructors and teaching assistants. (SC)

MATH V1102x or y Calculus II 3 pts. Prerequisites: MATH V1101 or the equivalent. Methods of integration, applications of the integral, Taylor's theorem, infinite series. (SC)

MATH V1201x or y Calculus III 3 pts. Prerequisites: MATH V1101 with a grade of B or better or Math V1102, or the equivalent. Vectors in dimensions 2 and 3, complex numbers and the complex exponential function with applications to differential equations, Cramer's rule, vector-valued functions of one variable, scalar-valued functions of several variables, partial derivatives, gradients, surfaces, optimization, the method of Lagrange multipliers. (SC)

MATH V1202x or y Calculus IV 3 pts. Prerequisites: MATHV1102, V1201, or the equivalent. Multiple integrals, Taylor's formula in several variables, line and surface integrals, calculus of vector fields, Fourier series. (SC)

MATH V1207x-V1208y Honors Mathematics A-B 4 pts. Prerequisites: (see Courses for First-Year Students). The second term of this course may not be taken without the first. Multivariable calculus and linear algebra from a rigorous point of view. Recommended for mathematics majors. Fulfills the linear algebra requirement for the major. (SC) Recitation Section Required.

MATH V2000x An Introduction to higher Mathematics 3 pts. Introduction to understanding and writing mathematical proofs. Emphasis on precise thinking and the presentation of mathematical results, both in oral and in written form. Intended for students who are considering majoring in mathematics but wish additional training.

MATH BC2001x Perspectives in Mathematics 1 pt. Prerequisites: Some calculus or permission of the instructor.

MATH BC2006y Combinatorics 3 pts.Not offered in 2009-2010. Corequisites: MATH V2010 is helpful as corequisite, not required. Honors-level introductory course in enumerative combinatorics. Pigeonhole principle, binomial coefficients, permutations and combinations. Polya enumeration, inclusion-exclusion principle, generating functions and recurrence relations.

MATH V2010x or y Linear Algebra 3 pts. Prerequisites: V1201, or the equivalent. Matrices, vector spaces, linear transformations, eigenvalues and eigenvectors, canonical forms, applications. (SC)

MATH V2020x Honors Linear Algebra 3 pts. Prerequisites: Math V1201 A more extensive treatment of the material in Math V2010, with increased emphasis on proof. Not to be taken in addition to Math V2010 or Math V1207-V1208.

MATH V2500x or y Analysis and Optimization 3 pts. Prerequisites: Math V1102-Math V1201 or the equivalent and MATH V2010. Mathematical methods for economics. Quadratic forms, Hessian, implicit functions. Convex sets, convex functions. Optimization, constrained optimization, Kuhn-Tucker conditions. Elements of the calculus of variations and optimal control. (SC)

MATH V3007y Complex Variables 3 pts. Prerequisites: MATH V1202. An elementary course in functions of a complex variable. Fundamental properties of the complex numbers, differentiability, Cauchy-Riemann equations. Cauchy integral theorem. Taylor and Laurent series, poles, and essential singularities. Residue theorem and conformal mapping.(SC)

MATH V3020x Number Theory and Cryptography 3 pts. Prerequisites: one year of calculus. Congruences. Primitive roots. Quadratic residues. Contemporary applications.

MATH V3021y Combinatorial Number Theory 3 pts.Not offered in 2009-2010. Prerequisites: MATH V3020 or MATH W4041. Advanced topics in number theory. Continued fractions. Approximations by rational numbers. Transcendental numbers. Arithmetic functions. Partitions of numbers and their generating functions. Stress on the combinatorial and algorithmic aspects of number theory. Contemporary applications.

MATH V3025y Making, breaking codes 3 pts.Not offered in 2009-2010. Prerequisites: Calculus I, II, III and Linear Algebra. A concrete introduction to abstract algebra. Topics in abstract algebra used in cryptography and coding theory.

MATH V3027x Ordinary Differential Equations 3 pts. Prerequisites: MATH V1201 or the equivalent. Corequisites: MATH V2010. Equations of order one; systems of linear equations. Second-order equations. Series solutions at regular and singular points. Boundary value problems. Selected applications.

MATH V3028y Partial Differential Equations 3 pts. Prerequisites: MATH V3027 and MATH V2010 or the equivalent . Introduction to partial differential equations. First-order equations. Linear second-order equations; separation of variables, solution by series expansions. Boundary value problems.

MATH V3050y Discrete Time Models In Finance 3 pts. Prerequisites: MATH V1102, V1201(or V1101, V1102, V1201), V2010. Recommended: MATH V3027(or MATH E1210) and SIEO W3600. Elementary discrete time methods for pricing financial instruments, such as options. Notions of arbitrage, risk-neutral valuation, hedging, term-structure of interest rates.

MATH V3386x Differential Geometry 3 pts. Prerequisites: MATH V1202 or the equivalent. Local and global differential geometry of submanifolds of Euclidiean 3-space. Frenet formulas for curves. Various types of curvatures for curves and surfaces and their relations. The Gauss-Bonnet theorem.

MATH V3901x-V3902y Supervised Readings In Mathematics 2-3 pts. Prerequisites: the written permission of the staff member who agrees to act as sponsor (sponsorship limited to full-time instructors on the staff list), as well as the permission of the director of undergraduate studies. The written permission must be deposited with the director of undergraduate studies before registration is completed. Guided reading and study in mathematics. A student who wishes to undertake individual study under this program must present a specific project to a member of the staff and secure his or her willingness to act as sponsor. Written reports and periodic conferences with the instructor.

MATH V3951x-V3952y Undergraduate Seminars In Mathematics 3 pts. Prerequisites: two years of calculus, at least one year of additional mathematics courses, and the permission of the director of undergraduate studies. The subject matter is announced at the start of registration and is different in each section. Each student prepares talks to be given to the seminar, under the supervision of a faculty member or senior teaching fellow.

MATH V3997x-V3998y Supervised individual research 3 pts. Prerequisites: The written permission of the faculty member who agrees to act as a supervisor, and the permission of the director of the undergraduate studies. For specially selected mathematics majors, the opportunity to write a senior thesis on a problem in contemporary mathematics under the supervision of a faculty member. .

MATH W4007y Analytic Number Theory 3 pts. Prerequisites: Math V3007 A one semeser course covering the theory of modular forms, zeta functions, L -functions, and the Riemann hypothesis. Particular topics covered include the Riemann zeta function, the prime number theorem, Dirichlet characters, Dirichlet L-functions, Siegel zeros, prime number theorem for arithmetic progressions, SL (2, Z) and subgroups, quotients of the upper half-plane and cusps, modular forms, Fourier expansions of modular forms, Hecke operators, L-functions of modular forms.

MATH W4032y Fourier Analysis 3 pts. Prerequisites: three terms of calculus and linear algebra or four terms of calculus. Fourier series and integrals, discrete analogues, inversion and Poisson summation formulae, convolution. Heisenberg uncertainty principle. Stress on the application of Fourier analysis to a wide range of disciplines.

MATH W4041xy-W4042xy Introduction To Modern Algebra 3 pts. The second term of this course may not be taken without the first. Prerequisite: Math V1102-Math V1202 and MATH V2010, or the equivalent. Groups, homomorphisms, rings, ideals, fields, polynomials, field extensions, Galois theory.

MATH W4043y Advanced Topics In Algebra: Algebraic Number Theory 3 pts. Prerequisites: MATH W4041-W4042 or the equivalent. Algebraic number fields, unique factorization of ideals in the ring of algebraic integers in the field into prime ideals. Dirichlet unit theorem, finiteness of the class number, ramification. If time permits, p-adic numbers and Dedekind zeta function.

MATH W4044x Representations of Finite Groups 3 pts. Prerequisites: Math V2010 and Math W4041 or the equivalent. Finite groups acting on finite sets and finite dimensional vector spaces. Group characters. Relations with subgroups and factor groups. Arithmetic properties of character values. Applications to the theory of finite groups: Frobenius groups, Hall subgroups and solvable groups. Characters of the symmetric groups. Spherical functions on finite groups.

MATH W4045y Algebraic Curves 3 pts. Prerequisites: Mathematics W4041,W4042 and Mathematics V3007. Plane curves, affine and projective varieties, singularities, normalization, Riemann surfaces, divisors, linear systems, Riemann-Roch theorem.

MATH W4050y Topics In Geometry and Topology 3 pts.Not offered in 2009-2010. Prerequisites: Math W4041 Advanced topics in geometry and topology chosen by the instructor from the following list. Non-Euclidean geometry (e.g., hyperbolic, elliptic, projective), combinatorial topology, algebraic topology, knot theory, braid theory, Morse theory, dynamical systems, foliations, graph theory.

MATH W4051x Topology 3 pts. Prerequisites: MATH V1202, MATH V2010, and rudiments of group theory (e.g., MATH W4041). MATH V1208 or W4061 is recommended, but not required. Metric spaces, continuity, compactness, quotient spaces. The fundamental group of topological space. Examples from knot theory and surfaces. Covering spaces.

MATH W4052x Introduction to Knot Theory 3 pts. Prerequisites: Math V2010 or equivalent and Math W4041. Recommended: Math W4051 or equivalent. The study of algebraic and geometric properties of knots in R^3, including but not limited to knot projections and Reidemeister's theorm, Seifert surfaces, braids, tangles, knot polynomials, fundamental group of knot complements. Depending on time and student interest, we will discuss more advanced topics like knot concordance, relationship to 3-manifold topology, other algebraic knot invariants.

MATH W4053y Introduction to Algebraic Topology 3 pts.Not offered in 2009-2010. Prerequisites: MATH V21010, MATH W4041, MATH W4051 The study of topological spaces from algebraic properties, including the essentials of homology and the fundamental group. The Brouwer fixed point theorem. The homology of surfaces. Covering spaces.

MATH W4061xy-W4062xy Introduction To Modern Analysis 3 pts. Prerequisites: The second term of this course may not be taken without the first. Prerequisites: MATH V1202 or the equivalent and V2010. Real numbers, metric spaces, elements of general topology. Continuous and differential functions. Implicit functions. Integration; change of variables. Function spaces.

MATH W4065x Honors Complex Variables 3 pts. Prerequisites: MATH V1207 and Math V1208 or MATH W4061. A theoretical introduction to analytic functions. Holomorphic functions, harmonic functions, power series, Cauchy-Riemann equations, Cauchy's integral formula, poles, Laurent series, residue theorem. Other topics as time permits: elliptic functions, the gamma and zeta function, the Riemann mapping theorem, Riemann surfaces, Nevanlinna theory.

MATH W4071x Introduction To the Mathematics of Finance 3 pts. Prerequisites: MATH V1202, V3027, STAT W4150, SEIO W4150, or their equivalents. The mathematics of finance, principally the problem of pricing of derivative securities, developed using only calculus and basic probability. Topics include mathematical models for financial instruments, Brownian motion, normal and lognormal distributions, the BlackûScholes formula, and binomial models.

MATH W4081y Introduction To Differentiable Manifolds 3 pts. Prerequisites: MATH W4051 or W4061 and V2010. The implicit function theorem. Concept of a differentiable manifold. Tangent space and tangent bundle, vector fields, differentiable forms. Stoke's theorem, tensors. Introduction to Lie groups.

MATH W4155y Probability Theory 3 pts. Prerequisites: MATH W4061 or MATH V3007. A rigorous introduction to the concepts and methods of mathematical probability starting with basic notions and making use of combinatorial and analytic techniques. Generating functions. Convergence in probability and in distribution. Discrete probability spaces, recurrence and transience of random walks. Infinite models, proof of the law of large numbers and the central limit theorem. Markov chains.

MATH W4386x-W4387y Geometrical Concepts In Physics 3 pts.Not offered in 2009-2010. Prerequisites: MATH V1202 or the equivalent and V2010. Material from topology and differential geometry with illustrations of their use in electrodynamics, general relativity, and Yang-Mills theory. In particular topological and differential manifolds, tensors, vector bundles, connections, and Lie groups are covered.

MATH W4391x-W4392y Quantum Mechanics: An Introduction for Mathematicans and Physicists 3 pts. Prerequisites: Math V1202 or the equivalent and Math V2010. This course will focus on quantum mechanics, paying attention to both the underlying mathematical structures as well as their physical motivations and consequences. It is meant for undergraduates with no previous formal training in quantum theory. The measurement problem and issues of non-locality will be stressed.

MATH E1210x or y Ordinary Differential Equations 3 pts. Prerequisites: MATH V1201 or the equivalent. Special differential equations of order one. Linear differential equations with constant and variable coefficients. Systems of such equations. Transform and series solution techniques. Emphasis on applications.

APMA BC2001x Perspectives in Mathematics 1 pt. Prerequisites: Some calculus or permission of the instructor. Intended as an enrichment to the mathematics curriculum of the first two years, this course introduces a variety of mathematical topics (such as three dimensional geometry, probability, number theory) that are often not discussed until later, and explains some current applications of mathematics in the sciences, technology and economics.

MATH BC2001x Perspectives in Mathematics 1 pt. Prerequisites: Some calculus or permission of the instructor. Intended as an enrichment to the mathematics curriculum of the first two years, this course introduces a variety of mathematical topics (such as three dimensional geometry, probability, number theory) that are often not discussed until later, and explains some current applications of mathematics in the sciences, technology and economics.

APMA E4101y Introduction to nonlinear dynamical systems 3 pts. Prerequisites: MATH E1210 AND PHYS W3003. Introduction to nonlinear differential equations and dynamical systems and their chaotic behavior. Topics include Lagrangian and Hamiltonian formulation of mechanics, phase space dynamics. Critical points and stability. Systems of linear differential equations. Solving simple nonlinear differential equations. Onset of chaos. Systems with dissipation, limit points and cycles, strange attractors.

APMA E4101x Introduction to Dynamical Systems 3 pts. Prerequisites: APMA E2101 (or MATH E1210) and APMA E3101 An introduction to the analytic and geometric theory of dynamical systems; basic existence, uniqueness and parameter dependence of solutions to ordinary differential equations; constant coefficient and parametrically forced systems; Fundamental solutions; resonance; limit points, limit cycles and classification of flows in the plane (Poincare-Bendixson Therem); conservative and dissipative systems; linear and nonlinear stability analysis of equilibria and periodic solutions; stable and unstable manifolds; bifurcations, e.g. Andronov-Hopf; sensitive dependence and chaotic dynamics; selected applications.

APMA E4400y Introduction to biophysical modeling. 3 pts. Prerequisites: Advanced calculus or the instructor's approval. Introduction to physical and mathematical models of cellular and molecular biologoy. Physics at the cellular schale (viscosity, heat, diffusion, statistical mechanics). RNA transcription and regulation of genetic expression. Genetic and biochemical networks. Bioinformatics as applied to reverse-engineering of naturally-occurring networks and to forward-engineering of synthetic biological networks. Mathematical and physical aspects of functional genomics.

APMA E4901x Seminar: Problem in Applied Mathematics 1 pt. Required for all applied mathematics major in the junior year. Prerequisites or corequisites: Math V3027, V3028, and V2010, or their equivalents. Introductory seminars on problems and techniques in applied mathematics. Typical topics are nonlinear dynamics, scientific computation, economics, operation research, etc.

APMA E4903x Seminar: Problems In Applied Mathematics 3 pts. Prerequisites or corequisites: MATH V3007, V3028, and V2010, or their equivalents. Required for all applied mathematics majors in the senior year. It consists of the same weekly lecture as APMA E4901 plus two hours of tutorials per week. Examples of problem areas are nonlinear dynamics, asymptotics, approximation theory, numerical methods, etc.